AP®MathsCollege BoardCalculus BCStudy GuidesUnit 7: Differential EquationsUnit 7 OverviewUnit 7 Summary

Unit 7 Summary (College Board AP® Calculus BC): Study Guide

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 15 May 2026

Differential equations summary

Key definitions

A differential equation involves derivatives
The general solution is a family of curves that solve a differential equation
A particular solution is a single curve that solves a differential equation and satisfies an initial or boundary condition
A slope field is a diagram that shows the tangent of the general solution at different points
An exponential model satisfies $\frac{d y}{d t} = k y$ (growth) or $\frac{d y}{d t} = - k y$ (decay)
A logistic model satisfies $\frac{d y}{d t} = k y (a - y)$
- The limiting value (carrying capacity) is the value of $a$

Key formulas

Separation of variables can be used to solve $\frac{d y}{d x} = f (x) g (y)$ by solving $\int \frac{d y}{g (y)} = \int f (x) d x$
Euler's method for finding approximate solutions to $\frac{d y}{d x} = f' (x, y)$ uses the recursive formulas
- $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$
- $x_{n + 1} = x_{n} + ∆ x$
The solution to an exponential model is of the form $y = y_{0} e^{k t}$ (growth) or $y = y_{0} e^{- k t}$ (decay)
- $y_{0}$ is the initial value
The solution to a logistic model is of the form $y = \frac{a}{1 + A e^{- a k t}}$ ( $0 < a < y$ ) or $y = \frac{a}{1 - A e^{- a k t}}$ ( $y > a$ )

Key facts

The doubling-time or half-life of an exponential model is $t = \frac{\ln 2}{k}$
The rate of change of a logistic model is changing fastest when $y = \frac{a}{2}$
If a curve is concave up, then the approximation from Euler's method is an underestimate
If a curve is concave down, then the approximation from Euler's method is an overestimate

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